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Suppose, for each $n$, $X_n$ is a random variable over $\{0, 1/n,2/n,...,1 \}$ and consider a sequence of random variables $\{X_n \}$. Then, can we construct a subsequence $\{n'\}$ such that, for each $x \in [0,1]$, we have $Pr (X_n' =x) \rightarrow Pr (X=x) $ as $ n' \rightarrow \infty$?

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What is $X$? How is it distributed? Remember that if $x$ is irrational, $P(X_{n'}=x)=0$ for any sub sequence $\{n'\}$. –  Ashok Dec 26 '11 at 7:57

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Maybe (your question is not very clear), but the random variable $X$ could be unrelated to the sequence $(X_n)$. Assume for example that $\mathrm P(X_n=\frac{k}n)\leqslant c_n$ uniformly over $k$, where $c_n\to0$ (for example, $X_n$ could be uniform). Then, $\mathrm P(X_n=x)\to0$ for every $x$, hence the random variable $X$ is a solution if and only if the distribution of $X$ is atomless.

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